PARABOLA
1) Given the two ends of the letus rectum, the maximum number of parabola that can be drawn is
a) 1 b) 2 c) 0 d) infinite
2) If the focus of a parabola is (-2,1) and the directrix has the equation x + y= 3 then the vertex is
a) (0,3) b) (- 1, 1/2) c) (-1, 2) d) (2,-1)
3) if the vertex and the locus of a parabola are (-1,1) and (2,3) respectively then the equation of the directrix is
a) 3x + 2y +14=0
b) 3x + 2y -25 =0
c) 2x - 3y +10 =0 d) none
4) The vertex of a Parabola is (a,0) and the directrix is x + y = 3a. The equation of the parabola is
a) x²+ 2xy + y²+ 6ax + 10ay + 7a²= 0
b) x²- 2xy + y²+ 6ax + 10ay = 7a²
c) x²- 2xy + y²- 6ax + 10ay = 7a² d) none
5) If the vertex= (2,0) and the extremities of the latus rectum are (3,2) and (3,-2) then the equation of the parabola is
a) y²= 2x -4 b) x²= 4y - 8 c) y²= 4x - 8 d) none
6) Any point on the Parabola whose focus is (0,1) and the directrix is x + 2 = 0 is given by
a) (t²+1, 2t+1) b) (t²+1,2t+1) c) (t²,2t) d) (t²-1, 2t +1)
7) The equation of the parabolas whose vertex and focus are on the positive side of the x-axis at distances a and b respectively from the origin is
a) y²= 4(b - a)(x - a)
b) y²= 4(a - b)(x - b)
c) x²= 4(b - a)(y - a) d) none
8) The equation x²+ 4xy + 4y² - 3x - 6y - 4 = 0 represents a
a) circle b) parabola c) a pair of lines d) none
9) The equation λx²+ 4xy +4y² + λx + 3y +2 = 0 represents a Parabola if λ is
a) -4 b) 4 c) 0 d) none
10) The focus of the parabola y² - x -2y +2 = 0 is
a) (5/4, 1) b) (1/4,0 ) c) (1,1) d) none
11) The vertex of the parabola (y - a)²= 4ax(x + a) is
a) (-a,a) b) (a, -a) c) (-2a, 2a) d) (-a/2, a/2)
12) The equation of the axis of the parabola 9y² - 16x -12y -57 = 0 is
a) 2x = 3 b) y= 3 c) 3y= 2 d) x + 3y = 3
13) The length of the latus rectum of the parabola 169(x -1)² + (y-3)² = (5x -12y +17)² is
a) 14/13 b) 28/13 c) 12/13 d) none
14) The length of the latus rectum of the parabola x= ay²+ by + c is
a) a/4 b) a/3 c) 1/a d) 1/4a
15) The parametric equation of a parabola is x = t²+1, y= 2t +1. The Cartesian equation of this directrix is
a) x= 0 b) x +1 = 0 c) y= 0 d) none
16) If (2,-8) is at an end of a focal chord of the parabola y²= 32x then the other end of the chord is
a) (32,32) b) (32,-32) c) (- 2,8) d) none
17) A line L passing through the focus of the Parabola y²= 4(x -1) intersects the parabola in two distinct points. If m be the slope of the line L then
a) - 1< m < 1 b) m < -1 or m > 1 c) m ∈ R d) none
18) The HM of the segments of a focal chord of the Parabola y²= 4ax is
a) 4a b) 2a c) a d) a²
19) The length of a focal chord of the parabola y²= 4ax at a distance b from the vertex is c. Then
a) 2a²= bc b) a³= b²c c) ac = b² d) b²c = 4a³
20) The parabola y²= Kx makes an intercept of length 4 on the line x - 2y =1. then k is
a) (√105 -5)/10 b) (5 - √105)/10 c) (5 + √105)/10 d) none
21) A double ordinate of the parabola y½= 8px is of length 16p. The angle subtended by it at the vertex of the parabola is
a) π/4 b) π/2 c) π/3 d) none
22) The chord AB of the parabola y²= 4cx cuts the axis of the parabola at C. If A= (at₁², 2at₁), B(at₂², 2at₂) and AC: AB= 1:3 then
a) t₂= 2t₁ b) t₂ + 2t₁= 0 c) t₁ + 2t₂= 0 d) none
23) AB is a chord of the Parabola y²= 4ax. If its equation is y= mx + c and it subtends a right angle at the vertex of the parabola then
a) c= 4am b) a= 4mc c) c= - 4am d) a+ 4mc= 0
24) If t₁ and t₂ are the ends of a focal chord of the Parabola y²= 2x then
a) t₁½+ t₂²= 2 b) t₁ + t₂= 1 c) t₁t₂= 4 d) none
25) A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is (y-2)²= 4(x +1). After reflection, the ray must pass through the point
a) ( 0 ,2) b) (2,0) c) (0,-2) d) (-1,2)
26) The equation of a parabola is y²= 4x. P(1,3) and Q(1,1) are two points in the x- y plane . Then, for the parabola
a) P and Q are exterior points
b) P is an interior point while Q is an exterior point.
c) P and Q are interior points
d) P is an exterior point while Q is an interior point.
27) The point (a,2a) is an interior point of the region bounded by the parabola y²= 16x and the double ordinate through the focus. Then a belongs to the open interval
a) a< 4 b) 0< a < 4 c) 0< a < 2 d) a> 4
28) The ends of a line segment are P(1,3) and Q(1,1). R is a point on the line segment PQ such that PR : QR= 1: λ. If R is an interior point of the parabola y²= 4x then
a) λ ∈ (0,1) b) λ∈ (-3/5,1) c) λ∈ (1/2,3/5) d) none
29) The range of λ for which the point (λ,1) is exterior to both the parabola y²= |x| is
a) (0,1) b) (-1,1) c) (-1,0) d) none
30) The number of point with integral coordinates that lie in the interior of the region common to the circle x²+ y²= 16 and the parabola y²= 4x is
a) 8 b) 10 c) 16 d) none
31) The number of distinct real tangents that can be drawn from (0,-2) to the parabola y²= 4x is
a) one b) two c) zero d) none
32) The tangent to the parabola y²= 4x the points (1,2) and (4,4) meet on the line
a) x= 3 b) x + = 4 c) y = 3 d) none
33) The point of intersection of the tangents to the parabola y²= 4x at the points , where the perimeter t has the value 1 and 2, is
a) (3,8) b) (1,5) c) (2,3) d) (4,6)
34) The Triangle formed by the tangents to a parabola y²= 4ax at the ends of the latus rectum and the double ordinate through the focus is
a) equilateral b) isosceles c) right angle isosceles d) dependent on the value of a for its classification.
35) If two tangents drawn from the point (α,β) to the parabola y²= 4x be such that the slope of one tangent is double of the other than.
a) β= 2α²/9 b) α = 2β²/9 c) 2α = 9β² d) none
36) The tangent from the origin to the parabola y²+ 4 = 4x are inclined at
a) π/6 b) π/4 c) π/3 d) π/2
37) If y + b = m₁(x + a) and y+ b = m₂(x + a) are two tangents to the parabola y²= 4ax then
a) m₁ + m₂ = 0 b) m₁m₂ = 0 c) m₁m₂ = -1 d) none
38) The tangent to a parable at the vertex V and any point P meet at Q. If S be the focus then SP , SQ, SV are in
a) AP b) GP c) HP d) none
39) The equation of the common tangent to the equal parabolas y²= 4ax and x²= 4ay is
a) x + y + a =0 b) x + y =a c) x - y =a d) none
40) t₁ and t₂ ate two points on the parabola y²= 4x. If the chord joining them is a normal to the parabola at t₁ then
a) t₁ + t₂ = 0 b) t₁(t₁ + t₂) = 1 c) t₁(t₁ + t₂) +2 = 0 d) t₁ t₂ + 1= 0
41) The normal to the curve x= at², y= 2at at the point P(t) meet the curve again at Q(t'). Then t' is
a) t + 1/t b) -t - 2/t c) t + 2/t d) t - 1/t
42) The set of points on axis of the parabola y²= 4x +8 from which the 3 normals to the parabola are all real and different is
a) |(k,0)| k ≤ 2
b) |(k,0)| k > -2
c) |(0,k)| k > - 2 d) none
43) The number of distinct normals that can be drawn from (-2,1) to the parabola y²- 4x - 2y - 3 =0 is
a) 1 b) 2 c) 3 d) 0
44) If the line y= x + k is a normal to the Parabola y²= 4x then k can have the value
a) 2√2 b) 4 c) -3 d) 3
45) The arithmatic mean of the ordinates of the feet of the normal from (3,5) to the parabola y²= 8x is
a) 4 b) 0 c) 8 d) none
46) The area of the triangle formed by the tangent and the normal to the parable y²= 4ax, both drawn at the same end of the latus rectum, and the axis of the parabola is
a) 2√2a² b) 2a² c) 4a² d) none
47) If two of 3 feet of normals drawn a point to the parabola y²= 4x be (1,2) and (1,-2) then the third foot is
a) (2, 2√2) b) (2, -2√2) c) (0,0) d) none
48) Let P, Q, R be three points on a Parabola, normals at which are concurrent. The centroid of the ∆ PQR must line on
a) a line parallel to the directrix
b) the axis of the parabola
c) a line of slope 1 passing through the vertex d) none
49) The vertex of the parabola y²= 8x is at the centre of a circle and the parabola cuts the circle at the ends of its latus rectum. Then the equation of the circle is
a) x²+ y²= 4
b) x²+ y²= 20
c) x²+ y²= 80 d) none
50) The length of the common part of the parabola 2y²= 3(x +1) and the circle x²+ y² + 2x = 0 is
a) √3 b) 2√3 c) √3/2 d) none
51) The circle x²+ y² + 2λx = 0, x ∈R, touches the parabola y²= 4x externally. Then
a) λ> 0 b) λ < 0 c) λ> 1 d) none
52) The locus of the middle points of chords of a Parabola which subtends a right angle qthe vertex of the Parabola is
a) a circle b) an ellipse c) a Parabola d) none
53) The locus of a point which tangents to a parabola are right angles is a
a) straight line
b) pair of straight lines
c) circle d) Parabola
54) P is a point. Two tangents are drawn from it to the parabola y²= 4x such that the slope of one tangent is three times the slope of the other. The locus of P is
a) a straight line b) circle c) a Parabola d) an ellipse
55) The locus chords of the middle points of parallel chords of a Parabola x²= 4ay is a
a) straight line parallel to the x-axis
b) straight line parallel to the y-axis
c) circle
d) straight line parallel to a bisector the angles between the axes
56) The locus of the middle points of the chords of the parabola y²= 8x drawn through the vertex is a parable whose
a) focus is (2,0)
b) latus rectum= 8
c) focus is (0,2)
d) latus rectum= 4
57) The locus of the points of trisection of the double ordinates of a parabola is a
a) pair of lines
b) circle
c) parabola
d) straight line
B-2
58) The parabola x²+ 2x - 4y =0 has
a) vertex = (-1,-1)
b) latus rectum= 4
c) focus = (-1,3/4)
d) focus= (0,-1/4)
59) The equation of a parabola is 25({(x -2)²+ (y +5)²}= (3x + 4y -1)². For this Parabola
a) vertex = (2,-5)
b) focus= ( 2,-5)
c) directrix has the equation 3x + 4y -1=0
d) axis has the equation 3x + 4y -1=0
60) Let PQ be a chord of the Parabola y²= 4x. A circle drawn with PQ as a diameter passes through the vertex V of the parabola. If ar(∆ PVQ)= 20 unit² then the Co-ordinates of P are
a) (16,8) b) (16, -8) c) (-16,8) d) (-16, -8)
61) The equation of a tangent to the parabola y²= 9x from the point (4,10) is
a) x - 4y + 36=0
b) 81x - 8y - 162=0
c) 9x - 4y + 4=0
d) x - 4y - 36=0
62) If the tangents to the Parabola y²= 4ax at (x₁, y₁) , (x₂, y₂), cut at (x₃, y₃) then
a) x₁, x₃, x₂ are in AP
b) x₁, x₃, x₂ are in GP
c) y₁, y₃, y₂ are in AP
d) y₁, y₃, y₂ are in GP
63) The equation of a locus is y¹+ 2ax + 2by + c = 0. Then
a) it is an ellipse
b) it is a Parabola
c) its latus rectum= a
d) its latus rectum= 2a
64) A tangent to the parabola y²= 4ax is inclined at π/3 with the axis of the Parabola. The point of contact is
a) (a/3, -2a/√3)
b) (3a, -2√3a)
c) (3a, 2√3a)
d) (a/3, 2a/√3)
65) A chord PP' of a parabola cuts the axis of the Parabola at O. The feet of the perpendicular from P and P' on the axis are M and M' respectively.
a) AP b) GP c) HP d) none
66) Let the equation of a circle and a Parabola be x¹+ y² - 4x - 6=0 and y²= 9x respectively. Then
a) (1,-1) is a point on the common chord of contact
b) the equation of the common chord is y +1=0
c) the length of the common chord is 6 d) none
67) The equation the common tangent to the parabola y²= 2x and the circle x²+ y²+ 4x = 0 is
a) 2 √6 x + y = 12
b) x + 2 √6 y + 12 = 0
c) x - 2 √6 y + 12 = 0
d) 2 √6x + y = 12
68) Let there be two parabolas with the same axis, focus of each being exterior to the other and the latus rectum being 4a and 4n. The locus of the middle points of the intercepts between the Parabolas made on the lines parallel to the common axis is a
a) straight line if a= b
b) Parabola if a≠ b
c) Parabola for all, a, b
d) none
69) P is a point which moves in the x-axis plane such that point P is nearer to the centre of a square than of the sides. The four vertices of the square are (±a, ±a). the region in which P will move is bounded by parts of parabolas of which one has the equation
a) y²= a²+ 2ax
b) x²= a²+ 2ay
c) y² + 2ax= a² d)
1b 2c 3a 4b 5c 6d 7a 8c 9b 10a 11a 12c 13b 14c 15a 16a 17b 18b 19d 20a 21b 22b 23c 24c 25a 26d 27b 28a 29b 30a 31b 32c 33c 34c 35b 36d 37c 38b 39a 40c 41b 42d 43a 44c 45b 46c 47c 48b 49b 50a 51a 52c 53a 54c 55b 56d 57c 58bc 59bc 60ab 61ac 62 ab 63bc 64bd 65ad 66b 67ac 68bc 69ab 70 ABC
EXERCISE - B
1) 3[sin³(3π/2 - x) + sin³(3π+ x)] - 2[sin⁶(π/2+ x) + sin⁶(5π- x)] is equal to
a) 0 b) b) 1 c) 3 d) sin4x + sin6x e) none
2) Show that
a) sin⁶x + cos⁶x + 3 sin²x cos²x =1
b) 3(sinx - cosx)⁴+ 6(sinx + cosx)²+ 4(sin⁶+ cos⁶x) is independent of x.
c) (sin⁸x - cos⁸x)= (sin²x - cos²x)(1- 2 sin²x cos²x).
d) (3+ cos4x) cos2x = 4(cos⁸x - sin⁸x).
e) If sinx + cosx = a, then find the values of |sinx - cosx| and cos⁴x + sin⁴x).
f) If sinx + cosecx = 2, then sin²x + cosec²x is equal to 2. T/F
g) f(x)= cos²x + sec²x≥ 2. T/ F.
OR minimum value of f(x) is 2
h) Given A= sin²x + cos⁴x, then for all real x.
a) 1≤ A≤ 2 b) 3/4 ≤ A≤ 1 c) 13/16≤ A≤ 1 d) 3/4≤ A≤ 13/16
i) Let A= sin⁸x + cos¹⁴x, then for all real x
a) A≥ 1 b) 0< A≤ 1 c) 1/2 < A≤ 3/2 d) none
j) If x, y are acute, sinx = 1/2, cosy= 1/3, then (x + y) ∈
a) (π/3,π/2) b) (π/2, 2π/3) c) (2π/3, 5π/6) d) (5π/6,π)
k) sec⁴x(1- sin⁴x) - 2 tan²x = 1
l) tan²x - sin²x = sin⁴x sec²x= tan²x sin²x.
m) (cotA+ tanB)/(cotB + tanA)= cotA tanB.
n) (sinx + cosx)(tanx + cotx)= secx + cosecx.
o) (cosx cosecx - sinx secx)/(cosx + sinx)= cosecx - secx.
p) (1+ cotA- cosecA)(1+ tanA + secA)= 2.
q) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1
r) (tanx + secx -1)/(tanx - secx +1)= (1+ sinx)/cosx.
s) {cot²x(secx -1)/(1+ sinx)= sec²x {(1- sinx)/(1+ secx).
t) (secx +1- tanx)/(tanx - secx +1)= (1+ cosx)/sinx.
u) cosx/(1- tanx) + sinx/(1- cotx)= sinx + cosx.
EXERCISE - B
1) If tₙ = sinⁿx + cosⁿx, then (t₃ - t₅)/t₁ = (t₅ - t₇)t₃
2) Show that
a) tanx/(1- cotx) + cotx/(1- tanx)= secx cosecx +1.
b) (sinx + cosecx)½+ (cosx + secx)²= tan²x + cot²x +7.
c) (1+ cotx + tanx)(sinx - cosx)= secx/cosec²x - cosecx/sec²x.
d) (secx - cosecx)(1+ tanx + cotx)= tanx secx - cotx cosecx.
e) {2sinx tanx(1- tanx)+ 2 sinx sec²x}/(1+ tanx)² = 3sinx/(1+ tanx).
f) (tanx + cosecy)² - (coty - secx)²= 2 tanx coty (cosec x + secy)
g) {(1+ sinx - cosx)/(1+ sinx + cosx)}² = (1- cosx)/(1+ cosx).
h) If 2sinx/(1+ sinx - cosx) = y, then (1- cosx + sinx)/(1+ sinx) is also y.
i) {1/(sec²x - cos²x) + 1/(cosec²x - sin²x)} sin²x cos²x = (1- sin²x cos²x)/(2+ sin²xcos²x).
j) (cosecx - secx)(cotx - tanx)= (cosecx + secx)(secx cosecx -2)
3) If tanx + sinx = m and tanx - sin x = n, then show that m²- n²= 4 √(mn)
4) Eliminate x from the relations a secx = 1- b tanx ; a² sec²x = 5 + b² tan²x.
EXERCISE - 3
1) If cosecx - sinx = m, and secx - cosx = n, eliminate x
2) If cosecx - sinx = a³, secx - cosx = b³, then a²b²(a²+ b²)= 1.
3) If cotx + tanx = m, secx - cosx =n, eliminate x.
4) If c cos²x + 3c cosx sin²x = m, c sin³x + 3c cos²x sinx = n, then show that (m+ n)²⁾³ + (m - n)²⁾³ = 2c²⁾³.
5) If cosx+ sinx =√2 cosx , then show that cosx - sinx = √2 sinx.
6) If sinx + 5 cosx = 5, show that 5 sinx - 3 cosx = ± 3.
7) If a cosx + b sinx = p, a sinx - b cosx = q, show that a²+ b²= p²+ q².
8) If a cosx - b sinx = c, show that a sinx + b cosx = ±√(a²+ b²- c²).
9) If a sinx + b cosx = c, then show that (a - b tanx)/(b + a tanx)= ± √(a²+ b²- c²)/c.
10) If tan²(1- e²). Show that secx + tan³x coséx = (2- e²)³⁾².
11) If ax/cosθ + by/sinθ = (a²- b²), and ax sinθ/cos²θ - by cosθ/sin²θ = 0, show that (ax)²⁾³ + (by)²⁾³ = (a²- b²)²⁾³.
12) If sinθ = (m²- n²)/(m²+ n²), determine the values of tanθ, secθ, and cosecθ.
13) If tanθ = 2x(x +1)/(2x +1), determine sinθ and cosθ.
14) If cosθ = 2x/(1+ x²), find the values of tanθ and cosecθ.
15) If secx = p + 1/4p, then secx + tanx = 2p or 1/2p.
16) If secθ + tanθ = p, obtain the values of secθ, tanθ, sinθ in terms of p.
17) If cosx/cosy , sinx/siny= b, then (a²- b²) sin² y = a²- 1
18) If tanθ = p/q, show that (p sinθ - q cosθ)/(p sinθ + q cosθ) = (p²- q²)/(p²+ q²).
19) Is the equation sec²θ = 4xy/(x + y)² possible for real values of x and y ?
If not, then find out a relation between x and y so that it may be possible. x= y but x≠ 0
20) If m²+ n²+ 2mn cosθ = 1, and p²+ q²+ 2pq cosθ = 1, and mp + nq + (mq+ np) cosθ = 0, show that m²+ n²= cosec²θ.
1b 2)e) √(2- a²), 1- (1/2) (a²-1)² f) T g) T hb ib jb
EXERCISE - 4
1) The value of sin⁶θ+ cos⁶θ+ 3 sin²θ cos²θ is
a) 0 b) 1 c) 2 d) 3
2) The least value of 2 sin²θ+ 3 cos²θ is
a) 1 b) 2 c) 3 d) 5
3) The greatest value of sin⁴θ + cos⁴θ is
a) 1/2 b) 1 c) 2 d) 3
4) The value of sin²θ cos²θ(sec²θ + cosec²θ) is
a) 2 b) 4 c) 1 d) 0
5) If sinθ + cosecθ= 2, then sin²θ+ cosec²θ is equal to
a) 1 b) 4 c) 2 d) none
6) For how many values of x between 0 and 2π is the Equation 2 cosec2x citx - cot²x = valid?
a) 0 b) 2 c) 1 d) none
7) Incorrect statement is
a) sinθ= -1/5 b) cosθ= 1 c) secθ =1/2 d) tanθ= 20
8) sec²θ= 4xy/(x + y)² is true if and only if
a) x + y ≠ 0 b) x=y, x≠ 0 c) x= y d) x≠ 0, y ≠ 0
9) If x= a cos²θ sinθ and y= a sin²θ cosθ, then (x²+ y²)³/x²y² is independent of θ. T/F
10) The inequality ₂sin²θ + ₂cos²θ≥ 2√2 holds for all real θ T/F
11) The equation sinθ = x + 1/x holds true for all real θ. T/F
12) The least value of tan²θ + cot²θ is _____.
13) The value of sinθ cosθ(tanθ + cotθ) is ______
14) If for real x, the equation x + 1/x = 2 cosθ holds, then cosθ= ____
15) If secθ - cotθ = q, then the value of cosecθ = _____
1b 2b 3b 4c 5c 6d 7c 8b 9t 10t 11f 12)2 13) 1 14) cosθ= ±1 15)
No comments:
Post a Comment